Why doesn't my image classification network get better with training?
The answer (as detailed quite thoroughly here and here) is that specifying

model.compile(optimizer = 'rmsprop', loss = 'binary_crossentropy', metrics=['accuracy'])


causes keras to guess, incorrectly, that because I am using binary_crossentropy for the loss function, that I would want to use binary_accuracy as the way of reporting the accuracy metrics. Apparently, one should specify that one wants the categorical_accuracy metrics if one, as I do, has more than two classes.